Square roots a better way

1
Square roots - a better way

• Examples

2

• Number (N) Guess (a) N/a a1 1/2
  (a (N/a)) successive changes
• 20

3

• Number (N) Guess (a) N/a a1 1/2
  (a (N/a)) successive changes
• 20 5.000000

4

• Number (N) Guess (a) N/a a1 1/2
  (a (N/a)) successive changes
• 20 5.000000 4.000000

5

• Number (N) Guess (a) N/a a1 1/2
  (a (N/a)) successive changes
• 20 5.000000 4.000000 4.500000

6

• Number (N) Guess (a) N/a a1 1/2 (a
  (N/a)) successive changes
• 20 5.000000 4.000000 4.500000
• 4.500000

7

• Number (N) Guess (a) N/a a1 1/2 (a
  (N/a)) successive changes
• 20 5.000000 4.000000 4.500000
• 4.500000 4.444444

8

• Number (N) Guess (a) N/a a1 1/2 (a
  (N/a)) successive changes
• 20 5.000000 4.000000 4.500000
• 4.500000 4.444444 4.472222

9

• Number (N) Guess (a) N/a a1 1/2 (a
  (N/a)) successive changes
• 20 5.000000 4.000000 4.500000
• 4.500000 4.444444 4.472222
  -0.027778

10

• Number (N) Guess (a) N/a a1 1/2 (a
  (N/a)) successive changes
• 20 5.000000 4.000000 4.500000
• 4.500000 4.444444 4.472222
  -0.027778
• 4.472222

11

• Number (N) Guess (a) N/a a1 1/2 (a
  (N/a)) successive changes
• 20 5.000000 4.000000 4.500000
• 4.500000 4.444444 4.472222
  -0.027778
• 4.472222 4.472050

12

• Number (N) Guess (a) N/a a1 1/2 (a
  (N/a)) successive changes
• 20 5.000000 4.000000 4.500000
• 4.500000 4.444444 4.472222
  -0.027778
• 4.472222 4.472050 4.472136

13

• Number (N) Guess (a) N/a a1 1/2 (a
  (N/a)) successive changes
• 20 5.000000 4.000000 4.500000
• 4.500000 4.444444 4.472222
  -0.027778
• 4.472222 4.472050 4.472136
  -0.000086

14

• Number (N) Guess (a) N/a a1 1/2 (a
  (N/a)) successive changes
• 20 5.000000 4.000000 4.500000
• 4.500000 4.444444 4.472222
  -0.027778
• 4.472222 4.472050 4.472136
  -0.000086
• 4.472136

15

• Number (N) Guess (a) N/a a1 1/2 (a
  (N/a)) successive changes
• 20 5.000000 4.000000 4.500000
• 4.500000 4.444444 4.472222
  -0.027778
• 4.472222 4.472050 4.472136
  -0.000086
• 4.472136 4.472136

16

• Number (N) Guess (a) N/a a1 1/2 (a
  (N/a)) successive changes
• 20 5.000000 4.000000 4.500000
• 4.500000 4.444444 4.472222
  -0.027778
• 4.472222 4.472050 4.472136
  -0.000086
• 4.472136 4.472136 4.472136

17

• Number (N) Guess (a) N/a a1 1/2 (a
  (N/a)) successive changes
• 20 5.000000 4.000000 4.500000
• 4.500000 4.444444 4.472222
  -0.027778
• 4.472222 4.472050 4.472136
  -0.000086
• 4.472136 4.472136 4.472136
  0.000000
• Value from calculator 4.472135955 4.472136 to
  six figures

18

• Number (N) Guess (a) N/a a1 1/2 (a
  (N/a)) successive changes
• 24 5.00000000000000 4.80000000000000

19

• Number (N) Guess (a) N/a a1 1/2 (a
  (N/a)) successive changes
• 24 5.00000000000000 4.80000000000000 4.9000000
  0000000

20

• Number (N) Guess (a) N/a a1 1/2 (a
  (N/a)) successive changes
• 24 5.00000000000000 4.80000000000000 4.9000000
  0000000
• 4.90000000000000

21

• Number (N) Guess (a) N/a a1 1/2 (a
  (N/a)) successive changes
• 24 5.00000000000000 4.80000000000000 4.9000000
  0000000
• 4.90000000000000 4.89795918367347

22

• Number (N) Guess (a) N/a a1 1/2 (a
  (N/a)) successive changes
• 24 5.00000000000000 4.80000000000000 4.9000000
  0000000
• 4.90000000000000 4.89795918367347 4.898979591
  83673

23

• Number (N) Guess (a) N/a a1 1/2 (a
  (N/a)) successive changes
• 24 5.00000000000000 4.80000000000000 4.9000000
  0000000
• 4.90000000000000 4.89795918367347 4.898979591
  83673 -0.00102040816327
• 4.89897959183673

24

• Number (N) Guess (a) N/a a1 1/2 (a
  (N/a)) successive changes
• 24 5.00000000000000 4.80000000000000 4.9000000
  0000000
• 4.90000000000000 4.89795918367347 4.898979591
  83673 -0.00102040816327
• 4.89897959183673 4.89897937929598

25

• Number (N) Guess (a) N/a a1 1/2 (a
  (N/a)) successive changes
• 24 5.00000000000000 4.80000000000000 4.9000000
  0000000
• 4.90000000000000 4.89795918367347 4.898979591
  83673 -0.00102040816327
• 4.89897959183673 4.89897937929598 4.898979485
  56636

26

• Number (N) Guess (a) N/a a1 1/2 (a
  (N/a)) successive changes
• 24 5.00000000000000 4.80000000000000 4.9000000
  0000000
• 4.90000000000000 4.89795918367347 4.898979591
  83673 -0.00102040816327
• 4.89897959183673 4.89897937929598 4.898979485
  56636 -0.00000010627038

27

• Number (N) Guess (a) N/a a1 1/2 (a
  (N/a)) successive changes
• 24 5.00000000000000 4.80000000000000 4.9000000
  0000000
• 4.90000000000000 4.89795918367347 4.898979591
  83673 -0.00102040816327
• 4.89897959183673 4.89897937929598 4.898979485
  56636 -0.00000010627038
• 4.89897948556636

28

• Number (N) Guess (a) N/a a1 1/2 (a
  (N/a)) successive changes
• 24 5.00000000000000 4.80000000000000 4.9000000
  0000000
• 4.90000000000000 4.89795918367347 4.898979591
  83673 -0.00102040816327
• 4.89897959183673 4.89897937929598 4.898979485
  56636 -0.00000010627038
• 4.89897948556636 4.89897948556635

29

• Number (N) Guess (a) N/a a1 1/2 (a
  (N/a)) successive changes
• 24 5.00000000000000 4.80000000000000 4.9000000
  0000000
• 4.90000000000000 4.89795918367347 4.898979591
  83673 -0.00102040816327
• 4.89897959183673 4.89897937929598 4.898979485
  56636 -0.00000010627038
• 4.89897948556636 4.89897948556635 4.898979485
  56636

30

• Number (N) Guess (a) N/a a1 1/2 (a
  (N/a)) successive changes
• 24 5.00000000000000 4.80000000000000 4.9000000
  0000000
• 4.90000000000000 4.89795918367347 4.898979591
  83673 -0.00102040816327
• 4.89897959183673 4.89897937929598 4.898979485
  56636 -0.00000010627038
• 4.89897948556636 4.89897948556635 4.898979485
  56636 0.00000000000000

31

• Number (N) Guess (a) N/a a1 1/2 (a
  (N/a)) successive changes
• 24 5.00000000000000 4.80000000000000 4.9000000
  0000000
• 4.90000000000000 4.89795918367347 4.898979591
  83673 -0.00102040816327
• 4.89897959183673 4.89897937929598 4.898979485
  56636 -0.00000010627038
• 4.89897948556636 4.89897948556635 4.898979485
  56636 0.00000000000000
• Calculator value
• 4.89897948556635(61963945681494118)
• So our method is correct to 14 decimal places
  after the third step in the calculation.

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